En este post ya calculamos los corchetes de Lie para los elementos de la base \{ \hat{e}_i\} = \{ \partial_r, \frac{1}{r} \partial_\theta, \frac{\csc \theta}{r} \partial_\varphi \}. Vamos a calcular ahora los coeficientes de conmutacion c_{ijk} (todas las r, \theta, \varphi que aparecen a continuación son \hat{r}, \hat{\theta}, \hat{\varphi}).

Como [\hat{e}_i,\hat{e}_j] = c_{ij}^{\phantom{ij}k} \hat{e}_k, entonces:

[\hat{e}_1,\hat{e}_2] = c_{12}^{\phantom{12}m} \hat{e}_m = -\frac{1}{r}\hat{e}_2 \rightarrow c_{r \theta}^{\phantom{r \theta} \theta} = -\frac{1}{r} = -c_{\theta r}^{\phantom{r \theta} \theta}

[\hat{e}_1,\hat{e}_3] = c_{13}^{\phantom{13}m} \hat{e}_m = -\frac{1}{r}\hat{e}_3 \rightarrow c_{r \varphi}^{\phantom{r \varphi} \varphi} = -\frac{1}{r} = -c_{\varphi r}^{\phantom{r \varphi} \varphi}

[\hat{e}_2,\hat{e}_3] = c_{23}^{\phantom{23}m} \hat{e}_m = -\frac{\cot \theta}{r}\hat{e}_3 \rightarrow c_{\theta \varphi}^{\phantom{\theta \varphi} \varphi} = -\frac{\cot \theta}{r} = -c_{\varphi \theta}^{\phantom{\varphi \theta} \varphi}

Vamos a calcular ahora los tres coeficientes de rotación de Ricci que, por tener indices covariantes diferentes, podrían no ser simétricos en la base ortonormal, cuando si lo son en una holonómica. Empezamos por los correspondientes a los Christoffel \Gamma^{\theta}_{\phantom{\theta} r \theta} = \Gamma^{\theta}_{\phantom{\theta} \theta r}:

\hat{\gamma}^{\theta}_{\phantom{\theta} r \theta} = \frac{1}{2} \eta^{\theta \theta} (c_{\theta r \theta} + c_{\theta \theta r} - c_{r \theta \theta}) = \frac{1}{2}(\frac{1}{r} + \frac{1}{r}) = \frac{1}{r}

c_{\theta r \theta} = \eta_{\theta i} c_{\theta r}^{\phantom{\theta r}i} = c_{\theta r}^{\phantom{\theta r} \theta} = \frac{1}{r}

c_{\theta \theta r} = \eta_{r i} c_{\theta \theta}^{\phantom{\theta \theta}i} = c_{\theta \theta}^{\phantom{\theta \theta} r} = 0

c_{r \theta \theta} = \eta_{\theta i} c_{r \theta}^{\phantom{r \theta}i} = c_{r \theta}^{\phantom{r \theta} \theta} = -\frac{1}{r}

\hat{\gamma}^{\theta}_{\phantom{\theta} \theta r} = \frac{1}{2} \eta^{\theta \theta} (c_{\theta \theta r} + c_{\theta r \theta} - c_{\theta r \theta}) = \frac{1}{2} c_{\theta \theta r} = 0

y vemos que, efectivamente, ahora no son simétricos. Seguimos con los correspondientes a \Gamma^{\varphi}_{\phantom{\varphi} r \varphi} = \Gamma^{\varphi}_{\phantom{\varphi} \varphi r}:

\hat{\gamma}^{\varphi}_{\phantom{\varphi} r \varphi} = \frac{1}{2} \eta^{\varphi \varphi} (c_{\varphi r \varphi} + c_{\varphi \varphi r} - c_{r \varphi \varphi}) = \frac{1}{2}(\frac{1}{r} + \frac{1}{r}) = \frac{1}{r}

c_{\varphi r \varphi} = \eta_{\varphi i} c_{\varphi r}^{\phantom{\varphi r}i} = c_{\varphi r}^{\phantom{\varphi r} \varphi} = \frac{1}{r}

c_{\varphi \varphi r} = \eta_{r i} c_{\varphi \varphi}^{\phantom{\varphi \varphi}i} = c_{\varphi \varphi}^{\phantom{\varphi \varphi} r} = 0

c_{r \varphi \varphi} = \eta_{\varphi i} c_{r \varphi}^{\phantom{r \varphi}i} = c_{r \varphi}^{\phantom{r \varphi} \varphi} = -\frac{1}{r}

\hat{\gamma}^{\varphi}_{\phantom{\varphi} \varphi r} = \frac{1}{2} \eta^{\varphi \varphi} (c_{\varphi \varphi r} + c_{\varphi r \varphi} - c_{\varphi r \varphi}) = \frac{1}{2} c_{\varphi \varphi r} = 0

Finalmente, los últimos que pierden su simetría son:

\hat{\gamma}^{\varphi}_{\phantom{\varphi} \theta \varphi} = \frac{1}{2} \eta^{\varphi \varphi} (c_{\varphi \theta \varphi} + c_{\varphi \varphi \theta} - c_{\theta \varphi \varphi}) = \frac{1}{2}(\frac{\cot \theta}{r} + \frac{\cot \theta}{r}) = \frac{\cot \theta}{r}

c_{\varphi \theta \varphi} = \eta_{\varphi i} c_{\varphi \theta}^{\phantom{\varphi \theta}i} = c_{\varphi \theta}^{\phantom{\varphi \theta} \varphi} = \frac{\cot \theta}{r}

c_{\varphi \varphi \theta} = \eta_{\theta i} c_{\varphi \varphi}^{\phantom{\varphi \varphi}i} = c_{\varphi \varphi}^{\phantom{\varphi \varphi} \theta} = 0

c_{\theta \varphi \varphi} = \eta_{\varphi i} c_{\theta \varphi}^{\phantom{\theta \varphi}i} = c_{\theta \varphi}^{\phantom{\theta \varphi} \varphi} = -\frac{\cot \theta}{r}

\hat{\gamma}^{\varphi}_{\phantom{\varphi} \varphi \theta} = \frac{1}{2} \eta^{\varphi \varphi} (c_{\varphi \varphi \theta} + c_{\varphi \theta \varphi} - c_{\varphi \theta \varphi}) = \frac{1}{2} c_{\varphi \varphi \theta} = 0

Existe una manera alternativa de realizar todos estos cálculos, que dejaremos para un futuro post, utilizando las formas de conexión, las ecuaciones de estructura de Cartan asumiendo torsión nula y la derivada exterior.

Anuncios